Abstract:
Let XX be a smooth algebraic variety over a field KK, and let Δ:X→X×XΔ:X→X×X be the diagonal
embedding. Then the cohomology sheaves of the complex LΔ∗Δ∗OXLΔ∗Δ∗OX are canonically identified with the sheaves of differential forms on XX. In particular, there is a spectral sequence
from the Hodge cohomology of XX to the hypercohomology of the complex LΔ∗Δ∗OXLΔ∗Δ∗OX . If the
characteristic of the base field KK is 00 or larger then dimXdimX, the complex LΔ∗Δ∗OXLΔ∗Δ∗OX is formal,
i.e. quasi-isomorphic to the direct sum of its cohomology sheaves. It follows that in this
case the above spectral sequence degenerates at the first page. It has been a longstanding
question whether this degeneration holds in any characteristic. I will explain a recent result
of Akhil Mathew showing that the analogous spectral sequence fails to degenerate for the
classifying stack of the finite group scheme μpμp over Fp. This easily yields an example of a
smooth projective projective variety X such that the spectral sequence does not degenerate.